Transmission coefficient. See

A more sophisticated reaction path approach is to replace in eq. (40) by 8jjq(sJj This is the essence of the approach taken by Garrett and Truhlar and generalized by Miller et al. and Skodje and Truhlar for polyatomic reactions. Truhlar and coworkers have proposed one-dimensional paths which deviate from the reaction path in order to compute accurate tunneling probabilities from which transmission coefficients (see below) are then used to correct their version of variational transition state theory, the so-called improved canonical variational [transition state] theory (ICVT) (also see below). 

AG = 9kcalmol increasing AG by 1.4 kcal mol decreases kysT by a factor of 10. Since quantum-mechanical elfects can modify the transmission coefficient (see section 16.3.4), it is preferable whenever possible to determine AG independently from the temperatiue dependence of the rate constant. 

The quality of the reaction coordinate chosen to explore the reaction process can be tested by means of the computation of the transmission coefficient (see Eq. 15.1) that accounts from the fraction of trajectories that recross a particular dividing surface back to the reactants. This transmission coefficient can be evaluated according to the positive flux formulation [94], assuming that the trajectory is initiated at the barrier top with forward momentum along the reaction coordinate ... 

An inverse correlation occurs between the experimental value < i.expi and the theoretical values of the standard rate constant k,caic when the latter is computed from Eq. (1) using the adiabatic value of the transmission coefficient(i.e.,/c= 1), the solvent-independent frequency factorv = A 77/i (see solid circles in Fig. 18), and the solvent dependence is taken into account only via continuum A., (het) values obtained from Eq. (3). 

The transmission coefficient k is approximately 1 for reactions in which there is substantial (>4kJ) electronic coupling between the reactants (adiabatic reactions). Ar is calculable if necessary but is usually approximated by Z, the effective collision frequency in solution, and assumed to be 10" M s. Thus it is possible in principle to calculate the rate constant of an outer-sphere redox reaction from a set of nonkinetic parameters, including molecular size, bond length, vibration frequency and solvent parameters (see inset). This represents a remarkable step. Not surprisingly, exchange reactions of the type... 

Figure 15. Calculated values of the transmission coefficient k plotted as a function of the solvent viscosity rf for four barrier frequencies a>b at 7 = 0.85. The squares denote the calculated results for to = 3 x 1012 s I, the asterisks denote results for to = 5 x 1012 s-1, the triangles denote results for to = 1013 s I and the circles denote results for a>b = 2 x 1013 s l. The solid lines are the best-fit curves with exponents a 0.72 for wb = 3 x 1012s l, a 0.58 for wb = 5 x 1012 s-1,a 0.22 for wb = 1013 s l, and a 0.045 for cob = 2 x 10I3s-. Note here that the barrier crossing rate becomes completely decoupled from the viscosity of the solvent at wb = 2 x 10l3s-1. The transmission coefficient k is obtained by using Eq. (326). Note here that the viscosity is calculated using the procedure given in Section X and is scaled by a2/ /mkBT, and a>b is scaled by t -1. For discussion, see the text. This figure has been taken from Ref. 170.

Let us in the following derive a relation between the actual transmission coefficient, gh, and the non-adiabatic coefficient, Kna, since it will show why gh > na and which part of the power spectrum for the solvent motion that is responsible for this. We see from Eq. (11.87) that... 

Since na is equal to the transmission coefficient when the solvent molecules do not respond to the motion in the reaction coordinate ( frozen solvent molecules), the right-hand side of the equation represents the dynamical response of the solvent. We see that,... 

Calculations of - reaction rates by the transition-state method and based on calculated - potential-energy surfaces refer to the potential-energy maximum at the saddle point, as this is the only point for which the requisite separability of transition-state coordinates may be assumed. The ratio of the number of assemblies of atoms that pass through to the products to the number of those that reach the saddle point from the reactants can be less than unity, and this fraction is the transmission coefficient , k. (There are also reactions, such as the gas-phase colligation of simple radicals, that do not require activation and which therefore do not involve a transition state.) See also - Gibbs energy of activation, - potential energy profile, - Poldnyi. 

Up to moderately high energy ( 179%) of the activation barrier for reactant product in the Are isomerization reaction, the fates of most trajectories can be predicted more accurately by Eq. (11) as the order of perturbation calculation increases, except just in the vicinity of the (approximate) stable invariant manifolds (e.g., see Eig. 5), and that the transmission coefficient K observed in the configurational space can also be reproduced by the dynamical propensity rule without any elaborate trajectory calculation (see Eig. 6). Our findings indicate that almost all observed deviations from unity of the conventional transmission coefficient k may be due to the choice of the reaction coordinate whenever the k arises from the recrossings, and most transitions in chemical... 

In summing up we see that the assumption of a much looser complex might have raised k by a factor of 8 to 20, depending on the temperature, while the uncertainty in the transmission coefficient could introduce another factor of 2. Considering the over-all defects of the theory we may thus estimate the uncertainties in the preexponential factors calculated in this way at about an average factor of 10. The agreement obtained here for Hs, a factor of 2, is thus well within these expected uncertainties. 

These theoretical considerations also gave a basis for the consideration of the optimal distance of discharge, which is a result of competition between the activation energy AG and the overlap of electronic wave functions of the initial and final states. The reaction site for outer-sphere electrochemical reactions is presumed to be separated from the electrode surface by a layer of solvent molecules (see, for instance, [129]). In consequence, the influence of imaging interactions on AGJ predicted by the Marcus equation is small, which explains why such interactions are neglected in many calculations. However, considerations of metal field penetration show that the reaction sites close to the electrode are not favored [128], though contributions to ks from more distant reaction sites will be diminished by a smaller transmission coefficient. If the reaction is strongly nonadiabatic, then the closest approach to the electrode is favorable. 

Where ka AG, and Kel represent the Boltzmann constant, the free energy barrier for the ET process, and the transmission coefficient, respectively. In the present system, the electronic coupling between the reactant and the product is sufficiently strong (.see bellow) so that Kel - 1. AG is represented by the electronic coupling matrix element, H12, and the reorganization free... 

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